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Unformatted text preview: Chapter 1. The Growth of Money ACTSC231 Mathematics of Finance Department of Statistics and Actuarial Science University of Waterloo Fall 2010 Instructor: Chengguo Weng C. Weng (c2weng@uwaterloo.ca) p. 1/3 1 Interest (p10) Monday has time value XBox investment opportunities theory. XBox time preference theory. (p10) Interest is the payment by a borrower to a lender (investor) in return for the use of the capital. XBox Interest may be paid annually, monthly, weekly, or daily...... = continuously. XBox The original sum invested (or borrowed) is the capital or principal . XBox The total repaid is the principal plus interest. C. Weng (c2weng@uwaterloo.ca) p. 2/3 1 Example 1.1. John lends Tom $10,000 for 5 years. Tom offers the following three options of repayments: Opt1 : Pay $10,500 at the end of 5 years. Opt2 : Pay $100 at the end of each year, and repay the principal after 5 years. Opt3 : Make payment of principal and interest of $2,100 at the end of each year. Which option is most favorable for John and which one is best for Tom? C. Weng (c2weng@uwaterloo.ca) p. 3/3 1 Accumulation Function (p11) Consider a single investment of $1 at time 0 without any withdrawal up to time t &gt; XBox At time t , the accumulated value (or accumulation ) of the investment is the total of the principal plus interest added. XBox Notation: a ( t ) is used to denote the accumulation at time t of $1 invested at time 0. RHD a ( t ) as a function of t is called accumulation function. RHD a (0) = 1 RHD if interest is positive, a ( t ) is an increasing function of t C. Weng (c2weng@uwaterloo.ca) p. 4/3 1 Amount Function (p11) We use A K ( t ) to denote the accumulated value at time t of $ K invested at time 0. XBox Is it true: A K ( t ) = K a ( t ) ? RHD It may not be true e.g. A bank account earns 3% if K 5000 , and 2.5% otherwise. XBox Unless stated otherwise, we generally assume A K ( t ) = K a ( t ) . C. Weng (c2weng@uwaterloo.ca) p. 5/3 1 Example 1 of accumulation functions (p12-13) e.g.1. a ( t ) = 1 + 0 . 05 t (simple interest). 5% of original capital is added each year, paid continuously. Graph of a ( t ) : C. Weng (c2weng@uwaterloo.ca) p. 6/3 1 Example 2 of accumulation functions (p12-13) e.g.2. a ( t ) = 1 + 0 . 05 t , XBox where t denotes the floor of t , i.e. the integer part of t . XBox interest is added only at the end of year Graph of a ( t ) : C. Weng (c2weng@uwaterloo.ca) p. 7/3 1 Example 3 of accumulation functions (p12-13) e.g.3. a ( t ) = 1 . 05 t (compound interest). XBox 5% of the accumulation (capital+interest earned) is added continually. Graph of a ( t ) : C. Weng (c2weng@uwaterloo.ca) p. 8/3 1 Simple Interest (p15) An investment accumulates with simple at rate i per time unit (e.g. one year) paid continuously if and only if a ( t ) = 1 + it , t ....
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